京都大学数理解析研究所

ＩＳＳＮ1880−2818

数理解析研究所講究録 1610

Ｔｈｅ ｓｅｃｏｎｄ Ｊａｐａｎｅｓｅ−Ａｕｓｔｒａｌｉａｎ Ｗｏｒｋｓｈｏｐ ｏｎ Ｒｅａｌ ａｎｄ Ｃｏｍｐｌｅｘ Ｓｉｎｇｕｌａｒｉｔｉｅｓ

京都大学数理解析研究所

2008 年 8 月

RIMS K6kyOroku Z6ZO

7he second lapanese-Australian Morkshop on Real and Complex Singularities

August, 2008

Research instztute for A4Cathematical Sczences

2klyoto Unzverszty, Kyoto, uLapan

This is a report of research done at the Research Institute for Mathematical Sciences, Kyoto University The papers contamed herem are m final form

and will not be submitted for pubhcation elsewhere

Preface

These 10 years a lot of joint researches have been done by Austrahan and Japanese smgularitists, in particular, in the field of Real Analytic Singulari- ties. Based on this connection, the First Austrahan-Japanese Workshop on Real and Complex Singularities was held at the Univertsity of Sydney, Australia, in September 2005 Followmg the first workshop, the Second Japanese-Australian workshop was held at Research Institute for Mathematical Sciences (RIMS), Ky- oto University, Japan, in November 2007 This volume is the Proceedmgs of the second workshop

The workshop was supported by Grant-in-Aid for Scientific Research (No 17204007), Japan Society for the Promotion of Science, and RIMS Project. We would not have been able to accept many mathematicians from overseas without their assistance, We would hke to take this opportumty to exress our gratitude to the mvestigator of the grant, Professor Shigenori Matsumoto, and the Research Institute There were 43 participants, mostly from Australia, Europe and Japan We would hke to thank also the participants for their cooperation

We are going to hold the third Austrahan-Japanese Workshop on Singularities in Australia next year (September 2009) We hope many mathematicians will participate in it from Japan and Australia, and also from other countnes

July 2008 Satoshi Koike

The second Japanese-Australian Workshop on Real and Complex Singularities

at Room 420

Research Institute of Mathematical Sciences (RIMS), Kyoto University orgamsed by

Satoshi Koike (Hyogo University of Teacher Education), Toshizumi Fukui (Saitama University),

Laurentiu Paunescu (Umverszty of Sydncy), and Adam Harris (Univorsity of New England)

URL http.1!www.rimath saitarna-u ac Jpllab.Jp/Fukui120071JARCS2.html

PROGRAMME

26th November (Monday) 9 30 - 10 30

10 45 - 11 45

Kentaro Sali (Hokkaido Umversity, Japan) Smgulanties of de Sitter horocyclzc surfaeps Shymchi Izumiya (Hokkaido University, Japan) Flat hghthke hypersurcface in Minkowski space 13 15 - 14 15 Adam Harris (Umversity of New England, Austraha)

Higher analytic cohomology and the Laplace-Beltrami operator near an isolated smgularity

14 45 — 15 45 Vyacheslav Sedykh (Russian State Gubkm University of Oil and Gas, Russia) The coexistence of real corank 1 smgularities

and the contact geometry of space cuxves 16 OO – 17 OO Goo Ishikawa (Hokkaido Umversity, Japan)

Local classification of varieties m the sympleetic spaee

27th November (TUesday)

9 30 - 10 30 Jim Ward (University of Sydney, Austraha) Lines on a cubic surface

10 45 - 11 45 Carles Bivi6eAusina (Umversitat Pohtecmca de Valencia, Spam) Mixed multiphcities of ideals and the effective computation of LoJasiewicz exponents of complex analytic maps

13 15 - 14 15 Alexander Esterov (Independent Umversity of Moscow and University of Toronto, Russia and Canada) Poincar6-Hopf mdices and Newton polyhedra

14 45 – 15 45 Mutsuo Oka (Tbkyo University of Science, Japan) Topology of lmear system of plane curves

16 OO - 17 OO Tzee-Char Kuo (University of Sydney, Australia) Arc Space, Morse Stability, Equi-singularities 18 30 - Conference dmner

28th November (Wednesday)

930- 10 30 Osamu Saeki (Kyushu Umversity, Japan)

Cobordism of Morse maps and its apphcation to map germs 10 45 – 11,45 Boldizsar Kalm6r (Kyushu Umversity, Japan)

Fold maps, immersions and cobordism

13 OO - 14 OO Minoru Yamamoto (Kurume National College of Technology, Japan) On pseudQimmersions of a surface into the plane

14 10 - 15 10 Ruibm Zhang (University of Sydney, Austraha) Representations of Quantum Groups and

Cohomology of Quantum Equivaxiant Vectoi Bundles 15 15 – FYee discussion at Shisendo and mght view of Kyoto

29th Nqovember (Thursday)

9 30 - 10,30 Takashi Nishimura (Yokohama National University, Japan)

An upper bound for multiphcities of cQrank one simple smgularities 10 45 – 11 45 Michael Eastwood (Umversity of Adelaide, Austraha)

Smgularities of semiconformal mappings 13 15 – 14 15 Alex Dimca (Universit6 de Nice, IFhrance)

Charactenstic and resonance varieties

14 45 - 15.45 Adam Parusniski (Universit6 d’Angers, IFli7ance)

Equivalence relations of two variable real analytic function germs 16 OO – 17 OO Laurentiu Paunescu (University of Sydney, Austraila)

Tame nonsmooth inverse mappmg theorems

List

of participants

Takao Akahori (University of Hyogo, Japan)

Carles Bivia Ausina (Universitat Pohtbcnica de Valbncia, Spain) Liang Chen (Northeast Normal Umversity, China)

Alexandru Dimca (Universit6 de Nice, Fhrance) Michael Eastwood (University of Adelaide, Australia) Alexander Esterov (Umversity of Toronto, Canada) Hiroshi IFhijimoto (University of Kobe, Japan) Takuo Fukuda (Nihon University, Japan) Tbshizumi Fukui (Saitama University, Japan) Adam Harris (Umversity of New England, Austraha) Jonathan Hillman (University of Sydney, Austraha) Goo Ishikawa (Hokkaido University, Japan)

Hirotada Ito (Kmki Umversity, Japan) Shuzo Izumi (Kinki University, Japan)

Shyuichi Izumiya (Hokkaido University, Japan) Boldizsar Kalm6r (Kyushu Umversity, Japan)

Masayuki Kawashima (Tokyo University of Science, Japan) YUmiko Kitagawa (Osaka City University, Japan)

K Kitazaki (Japan)

Satoshi Koike (Hyogo Umversity of Teacher Education, Japan) Tzee-Char Kuo (Umversity of Sydney, Austraha)

Takayasu Kuwata (Tbkyo Denki University, Japan) Kimio MiyaJima (Kagoshima Umversity, Japan)

Takashi Nishimura (Yokohama National University, Japan) Kumi Obata (Kinki University, Japan)

Mutsuo Oka (Tokyo University of Science, Japan) Adam Parusiriski (Universit6 d’Angers, IFhrance) Laurentiu Paunescu (University of Sydney, Australia) Osamu Saeki (Kyushu University, Japan)

Kentaro Saji (Hokkaido University, Japan) Kazuhiro Sakuma (Kinki Umversity, Japan)

Vyacheslav Sedykh (Russian State Gubkin University of Oil and Gas, Russia)

Masatomo Takahashi (Muroran Institute of Technology, Japan) Tadashi Takahashi (Kobe University, Japan)

Susumu Tanabe (Kumamoto University, Japan) Masamichi Takase (Shinshu University, Japan) James N Ward (University of Sydney, Australia) Tadayuki Watanabe (Kyoto University, Japan) YUuki Yamaguchi (Kinki University, Japan)

Minoru Yamamoto (Kurume National College of ’Ilechnology, Japan) Shoji Ybkura (Kagoshima University, Japan)

Ruibin Zhang (University of Sydney, Australia)

Ｔｈｅ ｓｅｃｏｎｄ Ｊａｐａｎｅｓｅ−Ａｕｓｔｒａｌｉａｎ Ｗｏｒｋｓｈｏｐ ｏｎ Ｒｅａｌ ａｎｄ Ｃｏｍｐｌｅｘ Ｓ ｉｎｇｕｌａｒＭｅｓ 第二回日豪実・複素特異点研究集会

ＲＩＭＳ研究集会報告集

2007年11月26日〜11月29日 研究代表者 小他 敏司（Ｓａｔｏｓｈｌ Ｋｏｌｋｅ）

副代表者 福井 敏純（Ｔｏｓｈｉｚｕｍｉ Ｆｕｋｕ1）

目 次

1Ｓｍｇｕｌａｒｉｔｉｅｓ ｏｆ Ｓｅｍｉｃｏｎｆｏｒｍａｌ Ｍａｐｐｍｇｓ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一1 Ｕ Ｂｒｅｔａｇｎｅ Ｏｃｃｉｄｅｎｔａｌｅ Ｐａｕｌ Ｂａｉｒｄ

ＵＡｄｅｌａｉｄｅ Ｍｉｃｈａｅｌ Ｅａｓｔｗｏｏｄ

2ＯＮ ＴＨＥ ＣＯＮＮＥＣＴＩＶＩＴＹ ＯＦ ＳＯＭＥ ＣＯＭＰＬＥＴＥ ＩＮＴＥＲＳＥＣＴＩＯＮＳ 一 一ｅ 一ｄ一一一 一一ｅｅ11

3

4

Ｕ Ｎｉｃｅ Ｓｏｐｈｉａ Ａｎｔｉｐｏｌｉｓ Ａｌｅｘａｎｄｒｕ Ｄｉｍｃａ

Ｏｒｉｅｎｔａｂｉｌｉｔｙ ｏｆ ｒｅａｌ ｒｅｓｕｌｔａｎｔ ｓｍｇｕｌａｎｔｉｅｓ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一18 Ｉｎｄｅｐｅｎｄｅｎｔ Ｕ Ｍｏｓｃｏｗ ／ Ｕ Ｔｏｒｏｎｔｏ

ＯＮ Ｉ ＮＪＥＣＴＩＶＩＴＹ ＯＦ ＴＡＭＥ ＭＡＰＰＩＮＧＳ

埼玉大・理（Ｓａｌｔａｍａ Ｕ）

Ｕ Ｓａｖｏｉｅ Ｕ Ｓｙｄｎｅｙ

Ａ1Ｅｓｔｅｒｏｖ

−ｐｔｅ一一一一一一Ｎ一一一一一一ｐｅｅｅｔ−ｅｎｅ一 ｅ−ｐ一一一一一一ｅ−ｅ一一ｔ−ｅｐ一一ｔ一一ｐｅ一一一一一一ｎ一一一29

福井 敏純（Ｔｏｓｈｉｚｕｍｌ ＦｕｋＵｉ）

Ｋｒｚｙｓｚｔｏ ｆ Ｋｕｒｄｙｋａ

Ｌａｕｒｅｎｔｉｕ Ｐａｕｎｅｓｃｕ

5ＡＲＥＭＡＲＫ ＯＮ∂一ＣＯＨＯＭＯＬＯＧＹ ＷＩＴＨ ＳＵＰＰＯＲＴＳ

ＩＮ ＴＨＥ ＣＯＭＰ：ＬＥＭＥＮＴ ＯＦ Ａ ＣＯＮＥ ＳＩＮＧＵＬＡＲＩＴＹ 一・・一・・一・ 一一一・一 一ｅ・ 一・・一一一・一…。・一32 ＵＮｅｗ Ｅｎｇｌａｎｄ Ａｄａｍ Ｈａｍｓ

Ｍａｓａｒｙｋ Ｕ Ｍａｒｔｍ Ｋｏｌ血

6Ｌｏｃａｌ ｃｌａｓｓｉｆｉｃａｔｉｏｎ ｏｆ ｖａｒｌｅｔｉｅｓ ｍ ｔｈｅ ｓｙｍｐｌｅｃｔｌｃ ｓｐａｃｅ…一一一一・一。一……一一ｔｅ一一ｔ−ｅｄ一…一一。一一・一一一一一38 北大・理学（Ｈｏｋｋａｌｄｏ Ｕ） 石川 剛郎（Ｇｏｏ Ｉｓｈｌｋａｗａ）

7Ｌｌｇｈｔｌｌｋｅ ｈｙｐｅｒｓｕｒｆａｃｅｓ ｍ Ｌｏｒｅｎｔｚ−Ｍｍｋｏｗｓｋ1ｓｐａｃｅ一一一一一一・・一一一一一一一・・一一一・一一51 北大・理学（Ｈｏｋｋａｌｄｏ Ｕ） 泉屋 周一（Ｓｈｙｕｌｃｈｌ Ｉｚｕｍｌｙａ）

ドダ

8ＰＯＮＴＲＹＡＧＩＮ−ＴＨＯＭ−ＳＺＵＣＳ ＴＹＰＥ ＣＯＮＳＴＲＵＣＴＩＯＮ ＦＯＲ ＮＯＮ・ＰＯＳＩＴＩＶＥ ＣＯＤＩＭＥＮＳＩＯＮＡＬ ＳＩＮＧＵＬＡＲ ＭＡＰＳ ＷＩＴＨ ＰＲＥＳＣＲＩＢＥＤ

ＳＩＮＧＵＬＡＲ ＦＩＢＥＲＳ 一・一・・…一一一一…一一一一一一一一一一一・一一・一…ｖ一一一一・・・一・一・一一・一一一一一…一・一・ｐ−ｅ一・一一……一・・一。……一一66

九大・数理学（Ｋｙｕｓｈｕ Ｕ） Ｂｏｌｄｌｚｓａｒ Ｋａｌｍａｒ

ｇ ＴＨＥ ＥＮＲＩＣＨＥＤ ＲＩＥＭＡＮＮ ＳＰＨＥＲＥ ＡＮＤ ＳＴＡＢＩＬＩＴＹ 一一一一一。一■ｖ・・一・ｄ・一一一・一。一一一80 ＵＳｙｄｎｅｙ Ｔｚｅｅ−Ｃｈａｒ Ｋｕｏ

〃 Ｌａｕｒｅｎｔｌｕ Ｐａｕｎｅｓｃｕ

10Ａｍｅｔｈｏｄ ｔｏ ｍｖｅｓｔｌｇａｔｅ Ｔ4（ノ）,ＴＬ（ｆ）ａｎｄ ｌｔｓ ａｐｐｌｌｃａｔｌｏｎｓ 一…。一一一一。一一一一一一一一・・一一…・。。一84 横浜国大・教育人間科学（Ｙｏｋｏｈａｍａ Ｎａｔ Ｕ）西村 尚史（Ｔａｋａｓｈｉ・Ｎｉｓｈｉｍｕｒａ）

一1一

1 1ＣＯＢＯＲＤＩＳＭ ＯＦ ＭＯＲＳＥ ＭＡＰＳ ＡＮＤ ＩＴＳ ＡＰＰＬＩＣＡＴＩＯＮ ＴＯ ＭＡＰ ＧＥＲＭＳ ｅｅ一 ｎ100 九大・数理学（Ｋｙｕｓｈｕ Ｕ） 佐伯 修（Ｏｓａｍｕ Ｓａｅｋ1）

1 2Ｋｏｅｎｄｅｎｎｋ ｔｙｐｅ ｔｈｅｏｒｅｍｓ ｆｏｒ ｆｒｏｎｔｓ 一一一ｅ−ｅｅｅ−ｅ一 一 一一一一一一一一一一一一一一一一一一一一一一ｅ一一一一一一一一一一一一 一一一一114 岐阜大・教育（ＧｌｆＵ Ｕ） 佐治 健太郎（Ｋｅｎｔａｒｏ ＳａＪ1） 1 3Ｏｎ ｐｓｅｕｄｏ−ｌｍｍｅｒｓｌｏｎｓ ｏｆ ａ ｓｕｒｆａｃｅ ｍｔｏ ｔｈｅ ｐｌａｎｅ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一121

久留米工業高専（Ｋｕｒｕｍｅ・Ｎａｔ Ｃｏｌｌ Ｔｅｃｈ） 山本 稔（Ｍｉｎｏｒｕ Ｙａｍａｍｏｔｏ）

一 11